Active Vibration Control of Boring Bar Vibrations

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Active Vibration Control of

Boring Bar Vibrations

Linus Andrén

Lars Håkansson

Department of Signal Processing

School of Engineering

Blekinge Institute of Technology

Research Report No 2004:07

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Active Vibration Control of Boring

Bar Vibrations

L. Andr´en and L. H˚

akansson

August, 2004

Research Report

Department of Signal Processing Blekinge Institute of Technology, Sweden

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Abstract

The boring operation is a cumbersome manufacturing process plagued by noise and vibration-related problems. A deep internal boring operation in a workpiece is a classic example of chatter-prone machining. The manufacturing industry today is facing tougher tolerances of product surfaces and a desire to process hard-to-cut materials; vibrations must thus be kept to a minimum. An increase in productivity is also interesting from a manufacturing point of view. Penetrating deep and narrow cavities require that the dimensions of the boring bar are long and slender. As a result, the boring bar is inclined to vibrate due to the limited dynamic stiffness. Vibration affects the surface finish, leads to severe noise in the workshop and may also reduce tool life.

This report presents an active control solution based on a standard boring bar with an embedded piezo ceramic actuator; this is placed in the area of the peak modal strain energy of the boring bar bending mode to be controlled. An accelerometer is also included in the design; this is mounted as close as possible to the cutting tool. Embedding the electronic parts not only protects them from the harsh environment in a lathe but also enable the design to be used on a general lathe as long as the mounting arrangements are relatively similar. Three diﬀerent algorithms have been tested in the control system. Since the excitation source of the original vibrations, i.e. the chip formation process cannot be observed directly, the algorithms must be constructed on the basis of a feedback approach. Experimental results from boring operations show that the vibration level can be reduced by 40 dB at the resonance frequency of a fundamental boring bar bending mode; several of its harmonics can also be reduced signiﬁcantly.

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Chapter 1 Introduction

The lathe is a very useful and versatile machine in the workshop, and it is able to perform a wide range of machining operations. A boring operation is a metal cutting operation that bores deep, precise holes in the workpiece. A boring bar is characterized by great length in comparison to its diameter. The boring bar is clamped at one end to a tool post or a revolver and has a cutting tool attached at the free end. The cutting tool is used to perform metal cutting in a bore or cavity of the workpiece. Since a boring bar is usually long and slender, it is inclined to vibrate. Deep internal boring of a workpiece is a classic example of chatter-prone machining. Performing metal cutting under vibrating conditions will yield unsatisfactory results in terms of the surface ﬁnish of a workpiece, tool life and undesirable noise levels. The boring bar vibration was investigated in [1] and [2]. The vibration may be characterized by a stochastic process with time varying statistical properties and with non-linear characteristics. [1]. In internal turning operations, the boring bar motion usually consists of components in both the cutting speed direction and the cutting depth direction [1, 2]. However, the motion of a boring bar during a continuous machining operation is generally greatest in the cutting speed direction; and is related to one of the bar’s two fundamental bending modes [1, 2]. A frequent result of this resonant motion is extremely high boring-bar vibration levels [1]. A typical boring operation is illustrated in Fig. 1.1.

A conventional countermeasure to the vibrations is to equip boring bars with a tuned vibration absorber. The absorber is tuned to the frequency range of the fundamental bending mode of the boring bar by adjusting the weight of the reactive mass and the stiffness and damping properties of the elastic element. This will reduce the vibration level during the cutting operation. Active vibration absorbers based on inertial-mass actuators have also been investigated [3]. Active and passive vibration absorbers can provide some relief and are most effective when placed near the tool-end of the bar [7]. Active vibration control of machine-tool vibration, however, comprises a number of different methods for the introduction of a control force to the boring bar. In [8] the approach was to use an active clamping house, i.e. to let the clamping of the boring bar be the secondary vibrating source; the results were good. A similar approach was proposed as early as 1975 in [9]. The method uses a pivoting boring bar and an electro-hydraulic servo system as

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The cutting

speed d irection

Workpiece

The feed direction

Boring bar

The cutting tool

The cutting depth

direction

Figure 1.1: A typical boring operation. an actuator; the results are promising.

This report presents a diﬀerent method for the introduction of secondary vibration in boring bars. Here the actuator is mounted in a milled space in a longitudinal direction below the centerline of the boring bar. When the actuator applies a load on the boring bar in its longitudinal direction due to the expansion of the actuator, the boring bar will bend and stretch. By introducing secondary anti-vibrations via the actuator applied bending moment on the boring bar, the original vibrations from the cutting process can be reduced [5, 6, 13].

A challenge is to incorporate electronic devices into the harsh environment of a lathe. An active vibration control application includes actuators and sensors in conjunction with a control system. The actuator and accelerometer must be protected from the metal chips and cutting fluids. One of the goals was to make the active control system applicable to a general lathe. Embedding the active parts, i.e. the actuator and accelerometer, will not only protect them from the surrounding environment but will also allow the design to be used on a general lathe provided that the mounting arrangement is relatively similar. Due to the recent development of piezo ceramic actuators, the technique can be embedded into a boring bar. Milling a space in the boring bar reduces the bending stiffness; with piezo ceramic actuator technology, however, the space can be kept small and the bending stiffness reduction is moderate. The motion of the boring bar usually has components in both the cutting speed and the cutting depth directions. However, the boring bar

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vibration is to a great extent dominated by the motion in the cutting speed direction [1, 2]. In [5, 6] active vibration control using one actuator in the cutting speed direction indicates that the use of one actuator is a satisfactory solution.

When an active boring bar is used, a suitable control algorithm is needed. The first requirement with respect to the algorithm is that it should be able to handle the non-stationary environments which a boring process gives rise to. Since it is not possible to dis-tinguish between the the original boring bar vibrations and the secondary anti-vibrations, the algorithm must be based on a feedback approach. It may also be advantageous to consider the forward path, i.e. the signal transfer from the adaptive filter to the error sensor. A simple proportional or P controller, which is widely used in control theory, may help satisfy the first two requirements, i.e. handle non-stationary environments and based on a feedback approach. Where variations in the forward path are considered, more advanced controllers may be preferred. An algorithm that meets all requirements is the feedback filtered-x lms algorithm. This algorithm has proved successful in both the active control of tool vibration in external longitudinal turning and the active control of boring bar vibration [5, 6, 13]. Both the feedback filtered-x LMS algorithm and the Internal Model Control (IMC) controller based on an adaptive control FIR filter and governed by the Filtered-x LMS algorithm have been tested. Internal model control causes a feedback controller to work as a feedforward equivalent, provided the estimate of the forward path matches the actual forward path.

This report deals with active vibration control in boring operations using three differ-ent control strategies and three differdiffer-ent active boring bar designs.

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Chapter 2 Materials and Methods

2.1 Experimental Setup

All the experiments have been carried out on a MAZAK 250 Quickturn lathe, see Fig. 2.1; this has 18.5 kW spindle power, a maximum machining diameter of 300 mm and 1007 mm between the centers. In order to save material, the cutting operations were performed as external turning operations, although a boring bar was used as a tool holder, see Fig. 2.2.

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Figure 2.2: The turning operation used in the experiments.

2.1.1 Measurement Equipment and Setup

Three different control algorithms were used in the active control measurements: an ordinary PID controller, a feedback filtered-x LMS algorithm, and an Internal Model Control (IMC) controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. A block diagram of the eFiltered-xperimental setup for the active vibration control in boring operations can be seen in Fig. 2.3. A signal conditioning unit is also included in the digital controller in order to be able to adjust the level of the input and output of the DSP. When using the filtered-x LMS algorithms, the transfer function of the signal chain D/A converter, amplifier, structural transfer path from the actuator to the accelerometer and an A/D converter must be estimated in an initial phase prior the active vibration control. The experimental setup for investigating the forward path is illustrated in the block diagram in Fig. 2.4. The accelerometer in the cutting speed direction was used when estimating the forward path. The following equipment was used in the measurements.

• 2 Br¨uel & Kjær 4374 accelerometers.

• 1 Br¨uel & Kjær NEXUS 2 channel conditioning ampliﬁer 2692. • TEAC RD-200T DAT recorder.

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Digital Controller HP signal analyzer ch1 ch2 DAT recorder ch1 ch3 Amplifier A = 10 Piezo actuator Boring Bar Accelerometer in the cutting depth direction bk4374 NEXUS Charge Amplifier Accelerometer in the cutting speed direction bk4374 NEXUS Charge Amplifier ch2

Figure 2.3: A block diagram describing the experimental setup for active vibration control. • Texas Instruments DSP TMS320C32.

• Active boring bars with an embedded piezo ceramic actuator, see section 2.2.

2.1.2 Cutting Data and Machining Parameters

The workpiece material in the cutting experiments was chromium molybdenum nickel steel. The diameter of the workpiece was large (< 200mm) to ensure that the workpiece vibrations were negligible. The workpiece material SS 2541-03, chromium molybdenum nickel steel, is a quenched and tempered steel. This material excites the machine-tool sys-tem with a narrow bandwidth in the cutting operation [27]. It facilitates the introduction of major narrow-banded tool vibration in a turning operation, resulting in a deterioration in surface ﬁnish and severe acoustic noise levels [1, 27].

The cutting tools used were standard 55◦diagonal inserts. These have a tool geometry

designated by the ISO code DNMG 150618-SL and with chip breaker geometry for medium roughing. The carbide grade was TN7015.

The cutting data was selected in order to produce significant tool vibrations. These resulted in an observable deterioration in the workpiece surface as well as severe acoustic noise. After a preliminary set of trials, a suitable combination of cutting data and tool geometry was selected, see Table 2.1. Cutting data set No. 1 was selected for the pro-duction of significant tool vibrations for evaluating the three different control algorithms used in active control of boring bar vibration. Cutting data set No. 1 was also selected to facilitate investigation of the influence of the actuator location in the boring bar on the vibration control performance in the metal cutting process. To vary the tool vibration

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Signal Source HP signal analyzer ch1 ch2 DAT recorder ch1 ch2 Amplifier A = 10 Piezo actuator (double P804.10) Boring Bar Accelerometer bk 4374 s/n 2243930 0.1418 pC/ms-2 NEXUS Charge Amplifier 1mV/ms-2 0.142 pC/ms-2

Figure 2.4: The block diagram describing the experimental setup for both oﬄine and online estimation of the forward path.

Cutting data Geometry Cutting speed depth of cut Feed

set _{v (m/min)} _{a (mm)} _{s (mm/rev)}

No. 1 DNMG 150608-SL 80 1.0 0.2-0.3

No. 2 DNMG 150608-SL 100-150 0.5-1.5 0.2

Table 2.1: Cutting data and tool geometry

level in a controlled manner, a low cutting speed was selected, i.e. just beyond the limit of build up of edge eﬀects. The initial feed rate was selected in accordance with the lower chip-breaking limit of the insert. In the cutting experiments the cutting depth was increased to the limit of the control of the active boring bar. The cutting depth was sub-sequently slightly reduced to the maximum machining depth where maintaining control was possible; the feed rate was then gradually increased to the limit of active control.

By using cutting data set No. 2, it was occasionally possible to perform the boring operation without any large vibrations. Under such circumstances, it was possible to record data that enabled online estimation of the forward path during continuous turning.

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2.2 Active Boring Bar

A boring bar is usually long and slender in order to facilitate metal cutting in the bore of a workpiece. The boring bar used in the experiments was based on standard WIDAX S40T PDUNR15 boring bars, see Fig. 2.5. The diameter of the boring bar was 40 mm and the length 300 mm; 100 mm is required for the clamping. The ovarhang part thus constitutes 200 mm.

Figure 2.5: A CAD model of the standard boring bar WIDAX S40T PDUNR15. To perform active vibration control, an actuator and accelerometer must be applied to the vibrating object. The environment in which active control of boring bar vibration is designed to operate is harsh. The actuator and accelerometer must be protected from cutting fluids and metal chips resulting from the cutting operation. One possibility is to embed and seal the electronic parts into the boring bar. Accelerometers are usually so small that incorporating them into the design will have negligible effect on the bending stiffness of the boring bar. The accelerometer was mounted 25 mm from the tool tip and senses the vibrations in the cutting speed direction. The actuator, on the other hand, must be sufficiently large to produce adequate secondary vibrational forces to enable a sufficient increase of the dynamic bending stiffness. There are several ways of mounting the actuator in the boring bar. Three different mounting locations for the actuator have been tested in real-life cutting experiments. The difference between the active boring bars

Cutting depth direction Cutting speed direction 40 mm 15 mm 15 mm 18.5 mm Actuator α

Figure 2.6: Boring bar cross section with embedded actuator and α is the actuator oﬀset angle.

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boring bar cross section radius which intersects the actuator center towards the reversed cutting depth direction. The actuator oﬀset angle is illustrated in Fig. 2.6 and is denoted α. In all three cases, the actuator was mounted in a longitudinal direction below the centerline of the boring bar and adjacent to the clamping. When the actuator expands in length, it applies a load to the boring bar in its longitudinal direction; as a result, the boring bar will bend and stretch. Secondary anti-vibrations may thus be introduced by the actuator applied bending moment in order to reduce the original boring bar vibration, excited by the chip formation process during continuous machining. A schematic ﬁgure of the active boring bar control system is shown in Fig. 2.7

Boring Bar Cutting speed direction Embedded actuator Workpiece Primary excitation introduced by the material deformation process

W

Feedback controller

Secondary excitation via active actuator

Figure 2.7: A schematic ﬁgure of the active boring bar control system.

The ﬁrst active boring bar design was based on an embedded actuator with 0◦ actuator

oﬀset angle.

The second active boring bar design was based on an embedded actuator with 15◦

actuator oﬀset angle.

The third active boring bar design was based on an embedded actuator with 30◦

actuator oﬀset angle.

2.3 Control Algorithms

Control is the process of causing a system variable to conform to some desired value known as a reference value [17]. A special class of control theory is feedback control, which is

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used in the active vibration control application discussed here. A block diagram of the elementary parts of feedback control is provided in Fig. 2.8. Feedback control can be

Reference

sensor Actuator Plant

Output sensor Disturbance Output Controller

Σ

Figure 2.8: Block diagram of an elementary feedback control system.

found in a wide range of products ranging from simple heating systems to very complex processes. The central component is the plant, which must be controlled in some way. An output sensor senses a variable that is designed to imitate a reference signal. The controller uses the information from the reference signal and the output of the plant to produce a control signal to the actuator. The actuator is the physical part of the control system that has direct control authority on the plant. A simple analogy is a heating system. Here the reference is the desired temperature in the room, and a radiator is used as the actuator. The plant would be the room and the output of the plant is the actual temperature of the room. The temperature of the room is easily controlled by switching the radiator on and oﬀ.

Feedback control is not a new science. In 1788 James Watt invented the centrifugal governor; this was the ﬁrst feedback device to attract the attention of the entire engi-neering community and be accepted internationally [23]. The earliest feedback device known, however, can be found among the works of Ktesibios, Philon and Heron from the Hellenistic period ca 300 B.C. For further reading on the history of feedback control, see [23].

The choice of control algorithms must be based on the application. The boring opera-tion is a process which has non-staopera-tionary stochastic properties [1]; the algorithm must be able to handle variations in the plant being controlled. Another important factor which has strong eﬀect the choice of control algorithm is that the excitation source, the chip formation process, cannot be observed directly. The accelerometer senses both the vibra-tions resulting from the cutting process and those induced by the actuator. The algorithm must thus based on a feedback approach. The forward path, which is always present in an active vibration control application [19] must also be considered. In the particular control problem under discussion, the forward path basically consists of an ampliﬁer, actuator and the structural transfer path in the boring bar, see Fig. 2.9. Strictly speakin, the forward path also includes D/A and A/D converters as well as an accelerometer. Three controller algorithms suitable for evaluating purposes with respect to the application

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un-Actuator

Amplifier Structural

transfer path Forward path

Figure 2.9: The physical components of the forward path in the boring bar vibration control system.

der discussion are the simple PID controller, the more advanced feedback filtered-x LMS algorithm and the Internal Model Control, IMC, controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. In the filtered-x LMS algorithm and the IMC-based controller the estimate of the forward path was a 40 coefficient FIR filter. Both the adaptive controllers used an adaptiver FIR filter with 35 weights .

2.3.1 PID Controller

The proportional integral derivative PID controller is well-known and is, for instance, widely accepted in the processing industry [24]. Originally, it was implemented using analog technology [24]. However, today almost all controllers are implemented in com-puters [24]. The digital PID controller can be viewed as an approximation of its analog counterpart.

In a boring operation, the plant or forward path may be observed by an accelerometer mounted on the boring bar. Since the goal is to reduce vibration, acceleration should be as small as possible. The acceleration signal is denoted e(t); the control signal to the actuator is denoted y(t). The PID controller in continuous time t can be written as [24]

y(t) = K e(t) + 1 Ti _t 0 e(τ )dτ + Td de(t) dt (2.1)

where K is the gain of the controller, Ti is the integration time and Td is the derivative

time. The proportional part of the controller sets the constant gain. The integral part in conjunction with a proportional part improves steady state properties; when combined with derivative control, it also improves the transient properties [17].

A discretized version of the analog PID controller can be approximated as [24]

y(n) = P (n) + I(n) + D(n) (2.2) where P (n) = Ke(n − 1) (2.3) I(n) = I(n − 1) + K FsTie(n − 1) (2.4) D(n) = FsTd FsTd+ ND(n − 1) − FsKTdN FsTd+ N e(n − 1) − e(n − 2) (2.5)

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where Fs is the sampling frequency and N is a high frequency gain limitation of the derivative part. e(n) is the resulting error; in active vibration control in a boring operation it is also the accelerometer signal. Fig. 2.10 shows a block diagram of a feedback control

Forward path C

Σ

x(n) = e(n-1) y(n) d(n) e(n) Unit delay z-1 y_C(n) K

Figure 2.10: Block diagram of a feedback control system based on a digital P controller.

system with a digital P controller. Here the box with the unit delay z−1 between e(n) and

e(n − 1) at the input to the controller indicated that the subject is a digital controller in a feedback control system.

2.3.2 LMS Algorithm

The least mean square LMS algorithm was developed by Widrow and Hoff in 1960. The least square approach provides a powerful approach to digital filtering in situations where a fixed, finite length filter is applicable. This approach has been widely used in many areas [20]. It is an important member of the family of stochastic gradient algorithms [18]. The LMS algorithm is very simple and has therefore been made the standard against which other adaptive filtering algorithms are benchmarked [18].

In the active control of vibration application discussed here, an LMS algorithm was used when estimating the forward path of the system. A forward path is always present in active control applications [19]; if the ﬁltered-x LMS algorithm is used as a controller, an estimate of the forward path is also needed. A block diagram of the forward path estimation using an LMS algorithm is shown in Fig. 2.11. The task of the LMS algorithm is to imitate a desired signal d(n) by letting an input signal x(n) pass an adaptive ﬁlter

wn(k) to produce an output y(n). The algorithm adjusts the weights wn(k) so that the

error signal e(n) is minimized in the mean square sense. The error signal can be written as e(n) = d(n) − y(n) = d(n) − L−1 l=0 wn(l)x(n − l), (2.6)

where the adaptive FIR ﬁlter has L coeﬃcients. The weights are updated on average in the negative direction of the gradient. The gradient estimate in the LMS algorithm

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Noise generator Adaptive filter w(n) LMS Forward path

Σ

−

+

x(n) y(n) d(n) e(n)

Figure 2.11: Block diagram of the forward path estimation using an LMS algorithm.

consists of the derivatives of the square error signal with respect to each of the weights of the adaptive ﬁlter

∂e2(n) ∂wn(k) = ∂ d(n) − L−1 l=0 x(n − l)wn(l) ₂ ∂wn(k) =−2x(n − k)e(n) (2.7)

for k = 0, 1, . . . , L − 1. The weights can now be updated in the negative direction of the gradient estimate as      wn+1(0) wn+1(1) .. . wn+1(L − 1)     =      wn(0) wn(1) .. . wn(L − 1)     + µ      x(n) x(n − 1) .. . x(n − L + 1)     e(n), (2.8)

To enable convergence in the mean square of the LMS algorithm the step size µ is usually selected according to the inequality [28]

0 < µ < 2

LE[x2(n)] (2.9)

where E[x2(n)] is the power of the input signal. In practice, the step size is selected less

than 5 % of the upper bound in Eq. 2.9 [19].

The LMS algorithm can be summarized in vector notation as [21]

y(n) = wT_nx(n) (2.10)

e(n) = d(n) − y(n) (2.11)

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Insufficient spectral excitation of the LMS algorithm when implemented in limit nu-merical precision may result in a divergence of the adaptive weights [19], e.g. a noiseless sinusoid as the reference signal to an adaptive filter with more than two filter weights may have this effect. In that case, the unconstrained weights may grow out of bound [19]. A solution to the problem is to incorporate a leakage factor γ into the algorithm. The leaky LMS algorithm is obtained if the weight adjustment algorithm, Eq. 2.12, in the LMS algorithm is replaced by [19]

w(n + 1) = γw(n) + µe(n)x(n), (2.13)

where 0 < γ < 1; this usually selected close to unity [19].

2.3.3 Filtered-x LMS Algorithm

When the adaptive filter is followed by a forward path, the conventional LMS algorithm must be modified in order to ensure convergence. In active control applications there is always a forward path present [19]; the forward path must thus be compensated for. Morgan proposed two approaches in 1980 where one of the suggestions later resulted in the filtered-x LMS algorithm [10]. The algorithm was independently derived by Widrow [11] for adaptive control and by Burgess [12] in an active noise control application, both in 1981. The input signal to an adaptive algorithm is usually denoted x and when it is also filtered, the name is straightforward. Another common name for the algorithm is filtered-reference LMS since the input signal to the algorithm is commonly known as the reference signal.

The ﬁltered-x LMS algorithm is illustrated by the block diagram in Fig. 2.12. To compensate for the forward path the input signal to the weight adjustment algorithm is ﬁltered by an estimate of the forward path.

Estimate of foward path C* Adaptive filter w(n) LMS Forward path C

Σ

x(n) y(n) d(n) e(n) y_C(n) x_C*(n)

Figure 2.12: Block diagram of the ﬁltered-x LMS algorithm.

As in the LMS, the filtered-x LMS is designed to minimize the mean square error. The error signal is produced by the summation of a desired signal d(n) and the output signal of the forward path C, i.e. the output signal of the adaptive filter filtered by the forward path C. The output of the adaptive filter is denoted y(n); after passing the forward path

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it is denoted yC(n). By assuming that the adaptive ﬁlter weights are time invariant, the error signal can be written approximately as

e(n) = d(n) + yC(n) ≈ d(n) + LC−1 lC=0 C∗(lC)y(n − lC) (2.14) = d(n) + LC−1 lC=0 C∗(lC) Lw−1 lw=0 wn(lw)x(n − lC− lw) (2.15) = d(n) + Lw−1 lw=0 wn(lw) LC−1 lC=0 C∗(lC)x(n − lC− lw) (2.16) = d(n) + Lw−1 lw=0 wn(lw)xC∗(n − lw) (2.17)

where LC is the length of the a FIR ﬁlter estimate of the forward path C*, xC∗(n) =

_L_C₋₁

lC=0 C∗(lC)x(n − lC) is the ﬁltered reference signal and Lw is the length of the

adap-tive filter. Thus, the gradient estimate in the filtered-x LMS algorithm is based on the derivative of the squared error with respect to the adaptive filter weights

∂e2(n) ∂wn(k) = ∂ d(n) + Lw−1 lw=0 wn(lw)xC∗(n − lw) ₂ ∂wn(k) = 2xC ∗(n − k)e(n) (2.18)

for k = 0, 1, . . . , L − 1. The weights can now be updated as      wn+1(0) wn+1(1) .. . wn+1(Lw − 1)     =      wn(0) wn(1) .. . wn(Lw− 1)     − µ      xC∗(n) xC∗(n − 1) .. . xC∗(n − Lw+ 1)     e(n) (2.19)

where µ is the step size or convergence factor. The ﬁltered-x LMS algorithm can be summarized in vector notations as

y(n) = wT(n)x(n), (2.20)

e(n) = d(n) + yC(n) (2.21)

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the ﬁltered reference signal vector is produced as x_C∗(n) =                LC−1 lC=0 C∗(lc)x(n − lC) LC−1 lC=0 C∗(lc)x(n − lC− 1) .. . LC−1 lC=0 C∗(lc)x(n − lC− Lw+ 1)                (2.23)

In order to select a step size µ to enable the ﬁltered-x LMS algorithm to converge, the following inequality is commonly used [25]

0 < µ < 2

E[x2_C∗(n)](Lw+ ∆)

(2.24)

where ∆ is the overall delay in the forward path in samples, Lw is the length of the

adaptive FIR ﬁlter and E[x2_C∗(n)] is the mean square value of the ﬁltered reference signal

to the algorithm.

One way to justify this algorithm is to consider what happens when the adaptive filter only changes slowly over time in comparison with the time duration of the forward path. Under these conditions, the estimate of the forward path and the adaptive filter commute. Since the output of the adaptive filter is sensed after the forward path, an ordinary LMS algorithm can be used provided that the reference signal to the weight adjustment algorithm is filtered by an estimate of the forward path. In practice, the filtered-x LMS algorithm is stable even if the control filter changes within the time scale associated with the dynamic response of the forward path [26].

The feedback filtered-x LMS algorithm is obtained from the filtered-x LMS algorithm by using the error signal as reference signal. In Fig. 2.13 a block diagram of the feedback filtered-x LMS algorithm is shown and this algorithm is given by [25]

y(n) = wT(n)x(n), x(n) = e(n − 1) (2.25)

e(n) = d(n) + yC(n) (2.26)

w(n + 1) = w(n) − µxC∗(n)e(n) (2.27)

Here, the reference signal vector x(n) = [e(n−1), . . . , e(n−Lw)]T is the delayed estimation

error signal vector, where e(n) is the error or, in this case, the accelerometer signal. For the purpose of convergence of the feedback ﬁltered-x LMS algorithm, the inequality in Eq. 2.24 may be used as guidance when selecting the initial step size µ [25]. However, the estimate of the power in the ﬁltered reference signal used in the upper bound of this inequality, Eq. 2.24, should be estimated prior to control [25]. Where the error signal is

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Σ

x(n) = e(n-1) y(n) d(n) e(n) Unit delay z-1 y_C(n) x_C*(n)

Figure 2.13: Block diagram of the feedback ﬁltered-x LMS algorithm.

used as input to the control system, the algorithm acts as a feedback controller. This will complicate the relation between the mean square error and the ﬁlter weights, i.e. the mean square error will not be a quadratic function of the ﬁlter weights [25].

Incorporating a leakage factor stabilizes the algorithm [19] and improves the robustness of the feedback control system [25]. The weight updating function for the leaky ﬁltered-x LMS algorithm is deﬁned as [19]

w_n+1 _{= γw}_n− µe(n)x_C∗(n), (2.28)

where 0 < γ < 1 and usually selected close to unity [19].

2.3.4 IMC Controller based on the Filtered-x LMS Algorithm

Internal Model Control, IMC, is a control structure that has been particularly popular in process control [24]. Extending a feedback control system with IMC theoretically enables a feedback system to be transformed into a feedforward equivalent if the estimate of the forward path is identical to the actual forward path. The algorithm generates its reference signal on the basis of the output of the adaptive ﬁlter and the error signal. Fig. 2.14 shows a block diagram of the IMC controller based on the ﬁltered-x LMS algorithm.

The search for a minimum is made in the mean square sense. The error signal is the

summation of the desired signal d(n) and the output of the forward path yC(n), hence

e(n) = d(n) + yC(n). (2.29)

If one assumes that the estimate of the forward path is identical to the actual one, i.e.

C∗ = C, the estimate of forward path output signal inﬂuencing the process yC∗(n) equals

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Σ

x(n) y(n) d(n) e(n) Unit delay z-1 y_c(n) Estimate of foward path C*

Σ

d*(n)

-+

+

y_c*(n) x_c*(n)

Figure 2.14: Block diagram of a IMC controller based on the ﬁltered-x LMS algorithm.

then

d∗(n) = e(n) − yC∗(n) = d(n) + yC(n) − yC∗(n) = d(n). (2.30)

If C∗ = C, the feedback system in Fig. 2.14 can be transformed into a feedforward

equivalent as in Fig. 2.15 [19].

The diﬀerence between the above and not using IMC is that the reference signal x(n) is x(n) = d∗(n − 1) = e(n − 1) − yC∗(n − 1) = e(n − 1) − LC−1 lC=0 C∗(lC)y(n − 1 − lC). (2.31)

The adaptive IMC controller algorithm based on the ﬁltered-x LMS is given by

y(n) = wT(n)x(n), x(n) = d∗(n − 1) (2.32)

e(n) = d(n) + yC(n) (2.33)

w(n + 1) = w(n) − µxC∗(n)e(n) (2.34)

Here, the reference signal vector d∗_{(n − 1) = [d}∗_{(n − 1), . . . , d}∗_{(n − L}_w)]T _{is the delayed}

synthesized desired signal vector where d∗(n) is the synthesized desired signal. For con-vergence of the adaptive IMC controller algorithm, the inequality in Eq. 2.24 may be used for guidance when selecting the initial step size µ [25].

Unfortunately, perfect estimates of the forward path are seldom encountered [22]. There are various reasons for this imperfection, e.g. the forward path response changes

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Adaptive filter w(n) LMS Forward path C

Σ

x(n) y(n) d(n) e(n) Unit delay z-1 y_c(n) Forward path C x_c*(n) Estimate of foward path C*

Figure 2.15: Block diagram of feedforward equivalent of the feedback system using the IMC controller based on the ﬁltered-x LMS algorithm.

during operation and with non-linearities in the forward path. This suggests that the feed-forward system in Fig. 2.15 is not a perfect match with the feedback system in Fig. 2.14. Imperfect plant response leads to an undesired feedback loop due to the diﬀerence between

yC(n) − yC∗(n). Looking at Fig. 2.14 and assuming for a moment that the signals are

deterministic and that there is a time invariant adaptive ﬁlter (for conscience reasons), the synthesized desired signal can, from a frequency perspective, be written as [22]

D∗(f ) = D(f ) +C(f ) − C∗(f ) Y (f ) (2.35)

where

Y (f ) = W (f )D∗(f )e−j2πf (2.36)

and the error signal can be written as

E(f ) = D(f ) + C(f )W (f )D∗(f )e−j2πf (2.37)

The closed loop frequency response function can then be established as

Hcl(f ) = E(f ) D(f ) = 1 + C(f )W (f )e−j2πf 1−_{C(f ) − C}∗_{(f )} _{W (f )e}−j2πf = 1 + C ∗_{(f )W (f )e}−j2πf 1−_{C(f ) − C}∗_{(f )} _{W (f )e}−j2πf (2.38)

The adaptation of the filter can be seen as a feedforward configuration with a residual feedback loop around the adaptive filter as a result of the imperfect estimate of the forward path [22]. As the degree of mismatch between the forward path and the estimate of the forward path increases, the stable region of the adaptive control system decreases [22]. One simple way of preventing the adaptive weights from becoming too large and improving the robustness of the feedback control system is to use a leakage term in the filtered-x LMS algorithm in accordance with Eq. 2.28 [22].

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2.4 Spectral Properties

It is often considered desirable to be able to view the content of a signal or system in the frequency domain. A spectrum shows how the power or energy of a signal is distributed versus frequency. The coherence function shows the correlation between two signals; this can also be interpreted as a measure of linearity [15]. The frequency response function shows the amplitude and phase of a system versus frequency.

2.4.1 Spectrum Estimation

A common estimator of the spectral content of a signal is the Welch spectrum estimator. The data record is divided into segments; these are allowed to overlap and are windowed prior to computing the periodogram [14].

The Welch spectrum estimate is obtained by averaging a number of periodograms. Each periodogram is based on segments of a time sequence x(n), each segment consisting of N samples. The original time sequence of data must be divided into data segments as follows

xl(n) = x(n + lD) where

n = 0, 1, . . . , N − 1 l = 0, 1, . . . , L − 1

where lD is the starting point for each periodogram and D is the overlapping increment. If D = N there is no overlap and if D = N/2 there is a 50% overlap between the consecutive

time sequences or data segments xl(n) and xl+1(n).

Dividing the original time sequence into data segments is equivalent to multiplying the time series by a window. The Welch method allows the use of arbitrary windows such as kaiser, hanning, ﬂattop, etc. Hence, each periodogram is based on a windowed sequence w(n)xl(n) where n = 0, 1, . . . N − 1.

The Welch power spectral density estimator P∗(fk) is given by

P_xx∗ (fk) = 1 FsLNU L−1 l=0 N −1 n=0 xl(n)w(n)e−j2πnk/N 2 , fk = k NFs (2.39)

where k = 0, . . . , N/2, L is the number of periodograms, N is the length of the

peri-odogram, Fs the sampling frequency and

U = 1 N N −1 n=0 w(n) 2

is the window-dependent normalization factor for power spectral density estimates. For lightly damped mechanical systems, the normalized bias error in a spectral density

estimate can be approximated, at the resonance frequencies fr using [15]

εb ≈ − 1 3 Be Br ₂ , (2.40)

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where Br is the half power bandwidth of the smallest resonance peak and Be is the window-dependent resolution bandwidth; the latter is deﬁned as [1]

Be = N −1 n=0 w2(n) _{N −1} n=0 w(n) 2Fs= 1 N N −1 k=0 |W (k)|2 W2(0) Fs,

where W (k) is the Fourier transformed window function w(n).

The normalized random error εr of the spectral estimator is dependent on the choice

of time window w(n) and the overlap between the periodograms D. εr can be expressed

as [14] εr P_xx∗ (fk) = 1 L 1 + 2 L−1 q=1 L − q L ρ(q) (2.41) where ρ(q) = N −1 n=0 w(n)w(n + qD) 2 N −1 n=0 w2(n) (2.42)

is the correlation between the periodograms with overlap D and window function w(n).

If the window is hanning and there is no overlap εr

Pxx∗ (fk)

= 1/√L, and if the common

50% overlap is used εr

P_xx∗ (fk)

≈ 1/1.89L/2 [14]. The equivalent number of

uncor-related periodograms Le used in the average to produce the spectrum estimate is given

by Le= L 1 + 2 L−1 q=1 L − q L ρ(q) (2.43)

where ρ(q) is deﬁned by Eq. 2.42. Using the equivalent number of uncorrelated peri-odograms, the normalized random error can be expressed as

εr

Pxx∗ (fk)

= 1/Le (2.44)

for all choices of overlap.

After preliminary trials, the following were selected: data length, data segment length N, number of periodograms L, number of equivalent uncorrelated periodograms Le, digital

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Parameter Value

Total record length, T 50 s

Data segment length, N 8192

Number of periodograms, L 291

Equivalent number of perodograms, Le 275

Digital window w(τ ) Hanning

Overlapping 50 %

Sampling rate, Fs 24000 Hz

Table 2.2: Spectral density estimation parameters

2.4.2 Coherence Function Estimation

The coherence function estimate is calculated from spectral estimates of the input and output of the system and shows the degree of linear dependency between two signals. The coherence function estimate is deﬁned as [15]

γ∗2xy(fk) =

| P∗

xy(fk)|2

P_xx∗ (fk)Pyy∗ (fk)

(2.45)

where Pxy∗ (fk) is the cross power spectral density estimate between input x and output

y, P_xx∗ (fk) and Pyy∗ (fk) are the power spectral density estimates of x and y respectively.

The coherence function satisﬁes the condition 0≤ γ∗2_xy_(f_k)≤ 1.

In practice, when the coherence function is greater than zero and less than unity, one or more of the following four conditions exists [16]

• Extraneous noise is present in the measurements.

• Resolution bias errors are present in the spectral estimates. • The system relating x(n) to y(n) is not linear.

• The output y(n) is due to other inputs than x(n).

Estimating the coherence between the potential reference signals and error signals of a possible active control application enables a preliminary estimate to be made of a maximum theoretical performance of a potential feedforward active control system [22]. Good coherence is thus important in active control applications.

The normalized random error of a coherence function estimate is [15]

εr γ∗2xy(fk) ≈ √ 21− γ_xy2 _(f_k) | γxy(fk)| √ Le (2.46)

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2.4.3 Frequency Response Function Estimation

The dynamic characteristics of a constant parameter linear system that is physically realizable and stable can be described by a frequency response function H(f ) [15]. The frequency response function is usually complex valued and is commonly shown by its

amplitude- and phase function. A frequency response function may be estimated in

accordance with Hxy∗ (fk) =

Pxy∗ (fk) P_xx∗ (fk)

(2.47)

where Pxy∗ (fk) is the cross spectral density of the input signal x and the output signal y,

and P_xx∗ (fk) is the auto spectral density of the input signal. The amplitude function is

|H∗

xy(fk)|; the phase function Θ∗xy(fk) is deﬁned as

Θ∗_xy_(f_k) = arg

H_xy∗ (fk)

(2.48) The random error in frequency response function estimates for the amplitude func-tion [15] are εr(|Hxy∗ (fk)|) ≈ 1− γ_xy2 _(f_k) γ_xy2 (fk)2Le (2.49)

and for the phase function [15] εr(|Θ∗xy(fk)|) ≈ arcsin

εr(|Hxy∗ (fk)|)

(2.50)

where Le is deﬁned by Eq. 2.43.

2.5 Nyquist Diagram

The Nyquist stability criterion is a well-known stability test. It relates the open-loop frequency response to the number of closed-loop poles of the system in the right half-plane in the laplacian domain [17]. For discrete time systems, the stability area in the z-plane is the unit disc instead of the left half plane in the laplacian domain. The Nyquist criterion is especially useful for determining the stability of a closed-loop system when the open-loop system is given. Stability is determined using the frequency response of a complex system, perhaps with one or more resonances, where the magnitude curve passes

one several times and/or the phase crosses 180◦ several times [17]. The criterion is also

very useful in dealing with open-loop systems, unstable systems, non-minimum phase systems and systems with pure delays.

A Nyquist diagram may be created in accordance with the following three steps [29] 1. Estimate the frequency response functions for all parts of the system.

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2. Determine the open-loop frequency response for the complete system.

3. Plot the real- versus the imaginary part of the open-loop frequency response for the

complete system in the complex plane for f ∈ [−Fs/2, Fs/2].

W(f) Σ C(f) -+ Σ Reference Controller Forward path Noise Output -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Re{Hol(f)} Im{H ol (f)} Critical point a) b)

Figure 2.16: A simple feedback system in a) and an example of a nyquist plot in b).

A simple feedback system is given in Fig. 2.16a), where W (f ) and C(f ) are stable

dynamic systems, e.g. controller and forward path. The open-loop Hol(f ) and closed-loop

Hcl(f ) transfer functions are

Hol(f ) = W (f )C(f ) (2.51)

Hcl(f ) = W (f )C(f )

1 + W (f )C(f ) (2.52)

By examining the closed-loop system Hcl(f ) it becomes obvious that if W (f )C(f ) = −1

or Hol(f ) = −1, at some frequency, then the response of the feedback control system

would become unbounded at this frequency due to a division by zero. According to the Nyquist stability criterion, the system is considered stable if the Nyquist plot does not

encircle the critical point (−1, 0) [24]. An example of a Nyquist plot of a stable system is

given in Fig. 2.16b).

When determining the open-loop system, several frequency response function esti-mates are normally included. When estimating each of the transfer functions it is impor-tant to produce estimates with a high level of accuracy in order to enable an accurate estimate of the open-loop response for the control system.

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Chapter 3 Results

Three different feedback controllers have been tested in the active control of boring bar vibration: two adaptive controllers based on the filtered-x LMS, algorithm and a time invariant digital P controller. The influence of the placement of the actuator in the active boring bars when it comes to perfomance in active control of boring bar vibration has been investigated. In addition, the properties of the forward path when the boring bar is not in contact with the workpiece and during continuous cutting operations have also been considered.

Time history records of typical boring bar vibration during continuous turning in both the cutting speed and cutting depth directions are shown in Fig. 3.1. The corresponding spectra of the time history records are shown in Fig. 3.2.

3.1 The Forward Path

Three different active boring bars have been designed, all with an actuator located adja-cent to the clamping below the adja-centerline and within the surface boundary of the boring bar. The difference between the designs is the actuator offset angle α, see Fig. 2.6 in section 2.2. The different designs introduce different forward paths in an active vibration control system. Fig. 3.3 shows offline frequency response function estimates of the forward path produced for the three different active boring bar designs when the boring bar is not in contact with the workpiece. Note that the first resonance peak of the forward path

with 0◦ actuator oﬀset angle of the active boring bar is lower both in frequency and

am-plitude as compared with the other two. The coherence function estimates corresponding to the three diﬀerent forward paths are presented in Figs. 3.4 a) - c). The coherence function estimates indicate that the forward path output signal may, to a great extent, be explained linearly from its input in all three cases.

In the adaptive controllers, the forward path is modeled as an FIR filter. The frequency response function estimate and the corresponding Fourier transformed FIR filter estimate of the forward path are plotted in the same diagram for each of the active boring bar designs, see Figs. 3.5 a) - c). In the subsequent results the active boring bar design 2 has not been involved as it demonstrated a significant deterioration in dynamic stiffness

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0 5 10 15 20 −4000 −2000 0 2000 4000 Time [ms] Acceleration [m/s 2 ] a) 0 5 10 15 20 −500 0 500 Acceleration [m/s 2 ] Time [ms] b)

Figure 3.1: Boring bar vibration as a function of time a) in the cutting speed direction and b) in the cutting depth direction. Workpiece material SS2541-03, feed rate s=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min, cutting tool DNMG 150806-SL, grade TN7015.

in the turning operation as compared to an unmodified standard boring bar. The 35 coefficients of the FIR filter estimate of the forward path for boring bar designs 1 and 3,

with actuator oﬀset angles of 0◦ and 30◦ are shown in Fig. 3.6

Occasionally it is possible to perform a boring operation where the vibration level is low. Under such circumstances, it is possible to estimate the forward path during boring to see whether the forward path characteristics have changed or not as a result of the changed boundary conditions at the tool tip. In Figs. 3.7 a) and b) both the offline and the online frequency response function estimates as well as the corresponding Fourier transformed FIR filter estimate of the forward path are plotted in the same diagram for each respective active boring bar design. Note in particular that there is a slight change in the first resonance peak for both versions of the boring bars when estimating the forward path during turning. When estimating the forward path during continuous cutting, the cutting operation will affect the estimation. The coherence functions of the online estimates are given in Fig. 3.8. The errors associated with the online estimation of coherence functions and frequency response functions are shown in Fig. 3.9.

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0 2000 4000 6000 8000 −30 −20 −10 0 10 20 30 40 50 60 Frequency [Hz] PSD [dB rel 1 (m/s 2 ) 2 /Hz] 0 2000 4000 6000 8000 −30 −20 −10 0 10 20 30 40 50 60 Frequency [Hz] PSD [dB rel 1 (m/s 2 ) 2 /Hz] a) b)

Figure 3.2: Power spectral densities of boring bar vibration a) in the cutting speed di-rection and b) in the cutting depth didi-rection. Workpiece material SS2541-03, feed rate s=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min, cutting tool DNMG 150806-SL, grade TN7015. 0 500 1000 1500 2000 2500 3000 3500 4000 −40 −20 0 20 40 H* xy (f) [m/s 2 /V] Frequency [Hz] a) α = 0o α = 15o α = 30o 0 500 1000 1500 2000 2500 3000 3500 4000 −π 0 π Phase [rad] Frequency [Hz] b) α = 0o α = 15o α = 30o

Figure 3.3: Frequency response function estimates of the forward path for the three different active boring bar designs with a different actuator offset angle α, a) magnitude functions and b) phase functions.

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0 1000 2000 3000 4000 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] Coherence γ * 2 xy (f) 0 1000 2000 3000 4000 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] Coherence γ * 2 xy (f) a) b) 0 1000 2000 3000 4000 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] Coherence γ * 2 xy (f) c)

Figure 3.4: Coherence function estimates between the signal to the actuator and the acceleration signal in the cutting speed direction of the diﬀerent active boring bar designs

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0 1000 2000 3000 4000 −60 −40 −20 |H* xy (f)| [dB] Frequency [Hz] Direct FRF FIR based 0 1000 2000 3000 4000 −π 0 π Phase [rad] Frequency [Hz] Direct FRF FIR based 0 1000 2000 3000 4000 −60 −40 −20 |H* xy (f)| [dB] Frequency [Hz] Direct FRF FIR based 0 1000 2000 3000 4000 −π 0 π Phase [rad] Frequency [Hz] Direct FRF FIR based a) b) 0 1000 2000 3000 4000 −60 −40 −20 |H* xy (f)| [dB] Frequency [Hz] Direct FRF FIR based 0 1000 2000 3000 4000 −π 0 π Phase [rad] Frequency [Hz] Direct FRF FIR based c)

Figure 3.5: Frequency response function estimates of the forward path when the boring bar is not in contact with the workpiece, oﬄine, and the Fourier transformed oﬄine FIR

ﬁlter estimate of the forward path used in the controller at a) 0◦ actuator oﬀset angle, b)

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0 10 20 30 40 −0.01 −0.005 0 0.005 0.01 n Forward path c*(n) 0 10 20 30 40 −0.01 −0.005 0 0.005 0.01 n Forward path c*(n) a) b)

Figure 3.6: The 35 coeﬃcient FIR ﬁlter estimate of the forward path of the boring bar

design 1 and 3 a) 0◦ actuator oﬀset angle and in b) 30◦ actuator oﬀset angle.

0 200 400 600 800 1000 −40 −20 0 20 |H* xy (f)| [dB rel m/s 2 /V] Frequency [Hz] online FRF offline FRF FIR based 0 200 400 600 800 1000 −π 0 π Phase [rad] Frequency [Hz] online FRF offline FRF FIR based 0 200 400 600 800 1000 −40 −20 0 20 |H* xy (f)| [dB rel m/s 2 /V] Frequency [Hz] online FRF offline FRF FIR based 0 200 400 600 800 1000 −π 0 π Phase [rad] Frequency [Hz] online FRF offline FRF FIR based a) b)

Figure 3.7: Frequency response function estimates of the forward path during a continuous cutting operation (online) and when the boring bar is not in contact with the workpiece (offline), and the Fourier transformed offline FIR filter estimate of the forward path used

in the controller at a) 0◦ actuator oﬀset angle and b) a 30◦ actuator oﬀset angle. The

online estimation of the forward path was produced using workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015. In a) feed rate s=0.2mm/rev, cutting depth a=1.5mm, cutting speed v=100m/min and in b) feed rate s=0.2mm/rev, cutting depth a=0.5mm, cutting speed v=150m/min.

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0 1000 2000 3000 4000 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] Coherence γ * 2 (f) xy 0 1000 2000 3000 4000 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] Coherence γ * 2 (f) xy a) b)

Figure 3.8: Coherence function estimate of the online estimation of the boring bar. In a)

active boring bar design 1 with 0◦ actuator oﬀset angle and in b) active boring bar design

3 with 30◦ actuator oﬀset angle.

0 1000 2000 3000 4000 0 0.05 0.1 ε r [ γ * 2 (f) ] xy Frequency [Hz] a) 0 1000 2000 3000 4000 0 0.05 0.1 ε r [ H* xy (f) ] Frequency [Hz] b) 0 1000 2000 3000 4000 0 0.2 0.4 0.6 ε r [ γ * 2 (f) ] xy Frequency [Hz] c) 0 1000 2000 3000 4000 0 0.2 0.4 0.6 ε r [ H* xy (f) ] Frequency [Hz] d)

Figure 3.9: The random error associated with the coherence function estimate εr[γ∗2xy(f )]

and the frequency response function estimate εr[Hxy∗ (f )] of the the online estimation

of boring bar design 1 and 3 with 0◦ and 30◦ actuator oﬀset angle respectively. In a)

εr[γ∗2xy(f )] active boring bar design 1 and in b) εr[Hxy∗ (f )] of the same boring bar. In c)

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3.2 Active Vibration Control

The results of the active control of boring bar vibration are illustrated as power spectral densities of boring bar vibration with and without active vibration control. In section 3.2.1, a performance comparison of the different active boring bar designs is made, and in section 3.2.2 the reduction achieved using different algorithms is demonstrated. Finally, the stability and robustness of the feedback controllers are addressed in section 3.2.3. Here, the Nyquist diagrams are given for the feedback control system for each of the three different feedback controllers. Also, the introduction of leakage in the adaptive algorithms as a measure to improve robustness as well as the cost of this measure are addressed in Nyquist diagrams and on power spectral densities of the boring bar vibration.

3.2.1 Boring Bar Comparison

Three diﬀerent active boring bar designs were tested. The actuator was mounted with

diﬀerent actuator oﬀset angles, see Fig. 2.6 in section 2.2. The boring bar with a 15◦

actuator offset angle demonstrated a significant deterioration in dynamic stiffness in the turning operation as compared to an unmodified standard boring bar. This reduction in dynamic stiffness was not observed in the other two designs. The performance of the

remaining two active boring bars with an actuator oﬀset angle of 0◦ and 30◦ was evaluated

using the feedback ﬁltered-x LMS algorithm. Fig. 3.10 shows the power spectral density of boring bar vibration with and without active vibration control using active boring bar

design 1 using the 0◦ actuator oﬀset angle. The power spectral densities of boring bar

vibration with and without active vibration control using active boring bar design 3 using

30◦ actuator oﬀset angle, are shown in Fig. 3.11. Figs. 3.10 and 3.11 also demonstrate

that the vibration level was not only suppressed in the cutting speed direction but also that signiﬁcant vibration reduction was found in the cutting depth direction.

3.2.2 Algorithm Comparison

Three algorithms were tested in the active control of boring bar vibration: the feedback filtered-x LMS algorithm, an Internal Model Control (IMC) controller based on an adap-tive FIR filter governed by the filtered-x LMS algorithm, and a time invariant digital P controller. The algorithms were compared using the active boring bar design which

per-formed best, namely design 1 with a 0◦ actuator oﬀset angle. Fig. 3.12 shows boring bar

vibration control results obtained using the feedback ﬁltered-x LMS algorithm. Fig. 3.13 shows the corresponding results obtained by using the IMC controller. Control results obtained using the P controller are illustrated in Fig. 3.14. Finally, a photograph of the surface of a machined workpiece is shown in Fig. 3.15 with and without active vibration control.

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400 600 800 1000 1200 1400 1600 −20 −10 0 10 20 30 40 50 60 Frequency [Hz] PSD [dB rel 1 (m/s 2 ) 2 /Hz] Active off Active on 400 600 800 1000 1200 1400 1600 −20 −10 0 10 20 30 40 50 60 Frequency [Hz] PSD [dB rel 1 (m/s 2 ) 2 /Hz] Active off Active on a) b)

Figure 3.10: Power spectral densities of boring bar vibration with and without active

vibration control using an active boring bar with 0◦ actuator oﬀset angle and the

feed-back ﬁltered-x LMS algorithm, a) cutting speed direction and b) cutting depth direction. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

vibration control using an active boring bar with 30◦ actuator oﬀset angle and the

feed-back ﬁltered-x LMS algorithm, a) cutting speed direction and b) cutting depth direction. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

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vibration control using an active boring bar with an actuator oﬀset angle of 0◦ and the

feedback ﬁltered-x LMS algorithm. a) cutting speed direction and b) cutting depth di-rection. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

vibration control using an active boring bar with a 0◦ actuator oﬀset angle and an

IMC-based adaptive controller, a) cutting speed direction and b) cutting depth direc-tion. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

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vibration control using an active boring bar with a 0◦ actuator oﬀset angle and a P

controller, a) cutting speed direction and b) cutting depth direction. Workpiece mate-rial SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

Figure 3.15: Photograph of a machined workpiece with and without active vibration control. The controller turned on to the right; there is no control to the left.

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3.2.3 Stability and Robustness of Feedback Controllers

The stability of a feedback control system requires that its open loop frequency response

Hol(f ) does not violate the closed loop stability requirements, i.e. the Nyquist stability

criterion [24]. A closed loop system is said to be stable if the polar plot of the open loop

frequency response Hol(f ) for the feedback control system does not enclose the (−1, 0)

point in the Nyquist diagram. The greater the shortest distance between the polar plot

and the (−1, 0) point, the more robust the feedback control system is with respect to

variation in forward path response and controller response. An estimate of the open loop response for a feedback control system may be produced based on the controller frequency response function and the forward path frequency response function [17]. The open loop frequency responses for the digital P controller were produced for the 6 different controller gains used for controlling of boring bar vibration. In the case of adaptive control, the adaptive FIR filter coefficients obtained after convergence were Fourier transformed to produce the corresponding open loop frequency responses. The forward path frequency response function was estimated both offline, when the boring bar was not in contact with the workpiece, and online, during a continuous cutting operation with a low bor-ing bar vibration level. The online estimate of the forward path was conducted durbor-ing continuous turning in workpiece material SS2541-03; with cutting tool DNMG 150806-SL and grade TN7015, the cutting parameters were feed rate s=0.2mm/rev, cutting depth a=1.5mm and cutting speed v=100m/min. However, observe that during continuous cut-ting with severe boring bar vibration levels the dynamic response of the clamped boring bar generally has pronounced non-linear properties [1].

Open loop frequency responses for the boring bar vibration control system were pro-duced using 6 different P controller gains and an offline estimate of the forward path. The Nyquist diagram in Fig. 3.16 shows the polar plots of these P controller gains based on open loop frequency responses. The corresponding magnitude and phase functions are shown in 3.17. Based on the 6 different P controller gains and an online estimate of the forward path, open loop frequency responses for the boring bar vibration control system were produced; these are shown in the Nyquist diagram in Fig. 3.18. The corre-sponding magnitude and phase functions are shown in 3.19. Fig. 3.20 illustrates power spectral densities of boring bar vibration with and without P control for the 6 different gain settings.

The adaptive control of boring bar vibration was carried out with and without a leakage factor in the adaptive weight update equation; the leakage factors γ = 0.9999 and γ = 0.999 were used. The Nyquist diagram in fig. 3.21 shows the polar plots of the open loop frequency responses based on the feedback filtered-x LMS algorithm with and without leakage and with an offline estimate of the forward path. The magnitude and phase functions corresponding to Fig. 3.21 are shown in Fig. 3.22. Using an online estimate of the forward path, the corresponding polar plots of the open loop frequency responses were produced; these are illustrated in the Nyquist diagram in Fig. 3.23. The corresponding magnitude and phase functions are shown in Fig. 3.24. Similarly, polar plots of open loop frequency responses based on the adaptive IMC controller with and

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−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Im{H* ol (f)} Re{H* ol(f)} K=−2 K=−4 K=−6 K=−3 K=−5 K=−7

Figure 3.16: Nyquist diagram for a boring bar vibration control system based on a P controller for 6 diﬀerent gain factors K. An oﬄine frequency response function estimate of

the forward path was used based on an active boring bar with a 0◦ actuator oﬀset angle.

without leakage and with an oﬄine estimate of the forward path as well as an online estimate of the forward path were produced. The Nyquist diagram in Fig. 3.25 shows the open loop frequency responses based on the IMC controller with and without leakage and with an oﬄine estimate of the forward path. The corresponding magnitude and phase functions are shown in Fig. 3.26. Finally, the open loop frequency responses produced for the IMC controller with and without leakage and with an online estimate of the forward path are plotted in the Nyquist diagram in Fig. 3.27; the corresponding magnitude and phase functions are shown in Fig. 3.28.

One way of increasing the robustness of the adaptive control algorithms is thus to incorporate a leakage factor to the adaptive weight update equation. Figs. 3.29 a) and b) show power spectral densities of boring bar vibration with and without feedback ﬁltered-x control using the leakage factors γ = 1, 0.9999, 0.999. The corresponding boring bar vibration spectra obtained with and without adaptive IMC control using the leakage factors γ = 1, 0.9999, 0.999 are shown in Figs. 3.30 a) and b). It is clear that the increase in robustness is made at the expense of degraded performance.

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0 500 1000 1500 2000 −60 −40 −20 0 K=−2 K=−3 K=−4 K=−5 K=−6 K=−7 |H* ol (f)| [dB] Frequency [Hz] 0 500 1000 1500 2000 −3π −2π −π 0 Phase [rad] Frequency [Hz] for all K

Figure 3.17: The estimated open loop frequency response for a boring bar vibration control system based on a P controller for 6 diﬀerent gain factors K. An oﬄine frequency response function estimate of the forward path was used based on an active boring bar

with a 0◦ actuator oﬀset angle.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Im{H* ol (f)} Re{H* ol(f)} K=−2 K=−4 K=−6 K=−3 K=−5 K=−7

Figure 3.18: Nyquist diagram for a boring bar vibration control system based on a P controller for 6 diﬀerent gain factors K. An online frequency response function estimate of

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0 500 1000 1500 2000 −60 −40 −20 0 K=−2 K=−3 K=−4 K=−5 K=−6 K=−7 |H* ol (f)| [dB] Frequency [Hz] 0 500 1000 1500 2000 −3π −2π −π 0 Phase [rad] Frequency [Hz] for all K

Figure 3.19: The estimated open loop frequency response for a boring bar vibration control system based on a P controller for 6 diﬀerent gain factors K. An online frequency response function estimate of the forward path was used based on an active boring bar

Active Vibration Control of Boring Bar Vibrations

Active Vibration Control of

Boring Bar Vibrations

Linus Andrén

Lars Håkansson

Department of Signal Processing

School of Engineering

Blekinge Institute of Technology

Blekinge Institute of Technology

Research Report No 2004:07

Active Vibration Control of Boring

Bar Vibrations

L. Andr´en and L. H˚

akansson

August, 2004

Contents

Chapter 1

Introduction

The cutting

speed d irection

Workpiece

The feed direction

Boring bar

The cutting tool

The cutting depth

direction

Chapter 2

Materials and Methods

2.1

Experimental Setup

2.1.1

Measurement Equipment and Setup

2.1.2

Cutting Data and Machining Parameters

2.2

Active Boring Bar

2.3

Control Algorithms

Σ

2.3.1

PID Controller

Σ

2.3.2

LMS Algorithm

Σ

−

+

2.3.3

Filtered-x LMS Algorithm

Σ

Σ

2.3.4

IMC Controller based on the Filtered-x LMS Algorithm

Σ

Σ

-+

+

+

Σ

2.4

Spectral Properties

2.4.1

Spectrum Estimation

2.4.2

Coherence Function Estimation

2.4.3

Frequency Response Function Estimation

2.5

Nyquist Diagram

Chapter 3

Results

3.1

The Forward Path

3.2

Active Vibration Control

3.2.1

Boring Bar Comparison

3.2.2

Algorithm Comparison

3.2.3